As somebody mentioned, some areas don't allow lead shot at all. Which was my original reason. I've shot a dozen or so deer and several coyotes with lead buckshot, and the hevi shot is a noticeable difference. The deer I've shot with lead will often run a 100 yards or more jumping fences and creeks with no problem. Sometimes you wonder if you hit them or not. Even with several hits through the lungs.
The Deer shot buy the same size hevi shot (00), often drop in thier tracks and if not rarely go more than a few feet. We've had several broadside pass through shots as far as 70 yards away. With the hevi shot, you know when you hit em. It really crushes them. When they are able to move, they walk, they don't run and they don't go far.
There have been several test done and the patterning aside, the hevi-shot usually out penatrates lead by 30-50%. It is more exspensive, but in a season I rarely shoot more than 10$ worth on deer and maybe 30$ worth for the yotes. With the price of gas and everything else it is a small exspense.
And the piece of mind it gives me when I shoot a deer and can tell that I hit it is worth it. I've recovered every deer I've shot with buckshot, but the sick feeling in the pit of my stomach while I look for 15mins or sometimes even an hour isn't any fun.
Lead will work and has for years, but Hevi-shot works better.IMHO